Abstract
Let M, N be any disjoint subsets of the vertex set of the graph G with ‖M‖=m and ‖N‖=n. We say that G ε C(m, n) if there is a cycle K in G such that M \(\subseteq\)VK and N∩VK=φ.
If g Is k-connected, then it is an old result of Dirac that G ε C(k, 0). It is easy to produce k-connected graphs which are not C(k+1, 0). Hence the best we can hope of an arbitrary k-connected graph is that it is C(k, 0). However if we restrict our attention to k-connected regular graphs we can improve on C(k, 0). Indeed two recent papers have shown that 3-connected cubic graphs are C(9, 0) but not C(10, 0). In addition the 3-connected cubic graphs which are C(9, 0) but not C(10, 0) have also been characterised. Some interesting open questions exist for k-connected regular graphs in general.
Further results regarding the relation between graphs which are C(m1, n1) and C(m2, n2) are discussed.
New results in all of the above areas are discussed and the three main methods of proof analysed.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
R.E.L. Aldred, C(m, n) properties in graphs, this volume.
G.A. Dirac, In abstrakten Graphen vorhandene vollstandige 4-Graphen und ihre Unterteilungen, Math. Nachr., 22, 1960, 61–85.
M.N. Ellingham, D.A. Holton and C.H.C. Little, Cycles through six vertices excluding one edge in 3-connected cubic graphs, University of Melbourne, Mathematics Research Report No. 11, 1982.
M.N. Ellingahm, D.A. Holton and C.H.C. Little, Cycles through ten vertices in 3-connected cubic graphs, University of Melbourne, Mathematics Research Report, No. 17. 1982.
A. Gardiner and D.A. Holton, Cycles with prescribed and proscribed vertices, University of Melbourne, Mathematics Research Report No. 6, 1981.
B. Grunbaum and H. Walther, Shortness exponent of graphs, J. Comb. Th., 14A, 1973, 364–385.
R. Halin, Zur Theorie der n-fach zusammenhangenden Graphen, Abh. Math. Sem Hamburg, 33, 1969, 133–164.
D.A. Holton, Cycles through specified vertices in k-connected regular graphs, Ars Comb., to appear.
D.A. Holton, B.D. McKay, M.D. Plummer and C. Thomassen, A nine point theorem for 3-connected graphs, Combinatorica, 2, 1982, 53–62.
D.A. Holton and M.D. Plummer, Cycles through proscribed and forbidden sets, Annals of Discrete Math., 16, 1982, 129–147.
A.K. Kelmans and M.V. Lomonosov, When m vertices in a k-connected graph cannot be walked round along a simple cycle, Discrete Math., 38, 1982, 317–322.
A.K. Kelmans and M.V. Lomonosov, On cycles through given vertices of a graph, Amer. Math. Soc Abstracts, No. 82T-05-245, 3, 1982, 255.
A.K. Kelmans and M.V. Lomonosov, A cubic 3-connected graph having no cycles through given 10 vertices has the "Petersen form", Amer. Math. Soc. Abstracts, No. 82T-05-260, 3, 1982, 283.
G.H.J. Meredith, Regular n-valent n-connected non-hamiltonian non-nedge colorable graphs, J. Comb. Th., 14B, 1973, 55–60.
D.M. Mesner and M.E. Watkins, Some theorems about n-vertex connected graphs, J. Math. Mech., 16, 1966, 321–326.
U.S.R. Murty, A simple proof of Perfect's Theorem, Ars Comb., to appear.
H. Perfect, Applications of Menger's Theorem, J. Math. Anal. Appl., 22, 1968, 96–111.
M.D. Plummer, On path properties versus connectivity I, Proc. 2nd S.E. Conf. on Combinatorics, Graph Theory and Computing, L.S.U., Baton Range, 1971, 458–472.
M.D. Plummer, On the (m+, n−) connectivity of 3-polytopes, Proc. 3rd. S.E. Conf. on Combinatorics, Graph Theory and Computing, F.A.U., Boca Raton, 1972, 393–408.
M.D. Plummer and E.L. Wilson, On cycles and connectivity in planar graphs, Canad. Math. Bull., 16, 1973, 283–288.
M.E. Watkins and D.M. Mesner, Cycles and connectivity in graphs, Canad. J. Math., 19, 1967, 1319–1328.
E.L. Wilson, R.L. Hemminger and M.D. Plummer, A family of path properties for graphs, Math. Ann., 197, 1972, 107–122.
Editor information
Rights and permissions
Copyright information
© 1983 Springer-Verlag
About this paper
Cite this paper
Holton, D.A. (1983). Cycles in graphs. In: Casse, L.R.A. (eds) Combinatorial Mathematics X. Lecture Notes in Mathematics, vol 1036. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0071507
Download citation
DOI: https://doi.org/10.1007/BFb0071507
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-12708-6
Online ISBN: 978-3-540-38694-0
eBook Packages: Springer Book Archive